91影视

It is claimed that the value of the population proportion is less than 0.4. The sample size (n) is equal to 50, and the level of significance is equal to 0.05.

The true population proportion is given to be equal to 0.25.

The following hypotheses are set up:

Null Hypothesis: The population proportion is equal to 0.4.

Symbolically,\({H_0}:p = 0.4\)

Alternative Hypothesis: The population proportion is less than 0.4.

Symbolically,\({H_0}:p < 0.4\)

The test is left-tailed.

The level of significance is given to be equal to 0.05.

Thus, the z-score for a left-tailed test with\(\alpha = 0.05\)is equal to -1.645.

The value of the z-score is converted to the sample proportion as follows:

\(\begin{array}{c}z = \frac{{\hat p - p}}{{\sqrt {\frac{{pq}}{n}} }}\\\hat p = p + z\sqrt {\frac{{pq}}{n}} \\ = 0.4 + \left( { - 1.645} \right)\sqrt {\frac{{\left( {0.4} \right)\left( {1 - 0.4} \right)}}{{50}}} \\ = 0.2860\end{array}\)

Thus, the value of the sample proportion \(\left( {\hat p} \right)\) is equal to 0.2860.

The test statistic is computed below:

\(\begin{array}{c}z = \frac{{\hat p - p}}{{\sqrt {\frac{{pq}}{n}} }}\\ = \frac{{0.286 - 0.25}}{{\sqrt {\frac{{\left( {0.25} \right)\left( {0.75} \right)}}{{50}}} }}\\ = 0.59\end{array}\)

Therefore, the required value of z-score is equal to 0.59.

The value of\(\beta \)is the area on the graph to the left of the computed z-score.

Thus,

\(\begin{array}{c}\beta = P\left( {z < 0.59} \right)\\ = 0.7224\end{array}\)

Thus,\(\beta = 0.7224\).

a.

Now, the value of the power of the test is to be determined.

\(\begin{array}{c}{\rm{Power}}\;{\rm{of}}\;{\rm{the}}\;{\rm{Test}} = 1 - \beta \\ = 1 - 0.7224\\ = 0.2776\end{array}\)

Thus, the power of the test is equal to 0.7224 when the true proportion is equal to 0.25

b.

The value of\(\beta = 0.2776\).

c.

The power equal to 0.7224 or 72.24% is high. It means that there is a 72.24% probability of making the correct conclusion of rejecting the null hypothesis when the true population proportion is equal to 0.25.